2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167845Mednykh proved that for any finite group G and any orientable surface S, there is a formula for #Hom(pi_1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur. Both of these proofs use character theory and an explicit presentation for π_1. These results have been reproven using quantum field theory. Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory.11 pages, 6 figures. v3 clarifies the dicussion of the Frobenius-Schur indicators and improves the argument in the quaternionic caseQuantum AlgebraGeometric Topology57R56; 20C05text